We study the Undecided-State Dynamics (USD), a fundamental consensus process in which each vertex holds one of k decided opinions or the undecided state. We consider both the gossip model and the population protocol model. Prior work established tight bounds on the consensus time of this process only for the regime k=O(n/(log⁡n)2) (for the population protocol model) and k = O((n/log n)1/3) (for the gossip model), often under restrictive assumptions on the initial configuration.

The discrete incremental voting process (DIV), introduced by Cooper, Radzik, and Shiraga [OPODIS '23], operates on an undirected graph where each node has an integer opinion. In one step a randomly selected node interacts with its randomly selected neighbor and changes its opinion by 1 towards the neighbor's opinion. The final consensus opinion has expectation equal to the degree-weighted average of the initial opinions. We show that for graphs with n nodes, conductance Φ, and the ratio of the average to smallest degree γ, if the maximal difference between initial opinions is K, then the expected convergence time is O(n (K log(Kn) + γn)/Φ2). This bound is essentially optimal for graphs of bounded expansion. We also show that for regular graphs, if the second largest eigenvalue (in absolute value) is o(1/log2 n) and K is o(n/log2 n), then w.h.p. DIV converges to the rounded initial average opinion.

Given a graph G , and an ordering π of its vertices, a permutation forest F ( G , π ) is a spanning forest of G whose components are obtained as follows. For each vertex v , connect v to its first neighbour w in G that appears after v in the ordering π . If we regard this edge ( v , w ) as directed forward, from v to w , then each vertex has at most one forward edge, and the components of F ( G , π ) are arborescences. The roots of the components formed by this process are those vertices of G with no forward edge in the ordering π . This paper shows that the permutation forests of the random graphs G n , p have a threshold for the emergence of a linear size component around p = 1 / n . In contrast to the w.h.p. emergence of a unique giant in G n , p , the permutation forest process has the unusual property that, with positive probability, a number of linear size components occur within the threshold window.

We consider 3-Majority, a probabilistic consensus dynamics on a complete graph with n vertices, each vertex starting with one of k initial opinions. At each discrete time step, a vertex u is chosen uniformly at random. The selected vertex u chooses three neighbors v1, v2, v3 uniformly at random with replacement and takes the majority opinion held by the three, where ties are broken in favor of the opinion of v3. The main quantity of interest is the consensus time, the number of steps required for all vertices to hold the same opinion. This asynchronous version turns out to be considerably harder to analyze than the synchronous version and so far results have only been obtained for k = 2. Even in the synchronous version the results for large k are far from tight. In this paper we prove that the consensus time is for all k. These are the first bounds for all k that are tight up to a polylogarithmic factor.

Abstract. In an influential article, Papadimitriou [ On selecting a satisfying truth assignment, in Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), 1991, pp. 163–169] proved that a local search algorithm called WalkSAT finds a satisfying assignment of a satisfiable 2-CNF with [Formula: see text] variables in [Formula: see text] expected time. Variants of the WalkSAT algorithm have become a mainstay of practical SAT solving (see, e.g., [Hoos and Stützle, J. Autom. Reason., 24 (2000), pp. 421–481]). In the present article, we analyze the expected running time of WalkSAT on random 2-SAT instances. Answering a question raised by Alekhnovich and Ben-Sasson [ SIAM J. Comput., 36 (2007) pp. 1248–1263], we show that WalkSAT runs in linear expected time for all clause/variable densities up to the random 2-SAT satisfiability threshold.

Abstract. We study the existence of a directed Hamilton cycle in random digraphs with [Formula: see text] edges where we condition on minimum in- and out-degree at least one. Denote such a random graph by [Formula: see text]. We prove that if [Formula: see text] then [Formula: see text]

We consider a type of pull voting suitable for discrete numeric opinions which can be compared on a linear scale, for example, 1 ("disagree strongly"), 2 ("disagree"), …, 5 ("agree strongly"). On observing the opinion of a random neighbour, a vertex changes its opinion incrementally towards the value of the neighbour’s opinion, if different. For opinions drawn from a set {1,2,…,k}, the opinion of the vertex would change by +1 if the opinion of the neighbour is larger, or by -1, if it is smaller. It is not clear how to predict the outcome of this process, but we observe that the total weight of the system, that is, the sum of the individual opinions of all vertices, is a martingale. This allows us analyse the outcome of the process on some classes of dense expanders such as complete graphs K_n and random graphs G_{n,p} for suitably large p. If the average of the original opinions satisfies i ≤ c ≤ i+1 for some integer i, then the asymptotic probability that opinion i wins is i+1-c, and the probability that opinion i+1 wins is c-i. With high probability, the winning opinion cannot be other than i or i+1. To contrast this, we show that for a path and opinions 0,1,2 arranged initially in non-decreasing order along the path, the outcome is very different. Any of the opinions can win with constant probability, provided that each of the two extreme opinions 0 and 2 is initially supported by a constant fraction of vertices.

We consider two simple asynchronous opinion dynamics on arbitrary graphs where every node u of the graph has an initial value ξu(0). In the first process, which we call the NodeModel, at each time step t ≥ 0, a random node u and a random sample of k of its neighbours υ1, υ2, ... , υk are selected. Then, u updates its current value ξu(t) to [EQUATION], where α ∈ (0, 1) and k ≥ 1 are parameters of the process. In the second process, called the EdgeModel, at each step a random pair of adjacent nodes (u, υ) is selected, and then node u updates its value equivalently to the NodeModel with k = 1 and υ as the selected neighbour.

We consider a type of pull voting suitable for discrete numeric opinions which can be compared on a linear scale, for example, 1 ('disagree strongly'), 2 ('disagree'), ..., 5 ('agree strongly'). On observing the opinion of a random neighbour, a vertex changes its opinion incrementally towards the value of the neighbour's opinion, if different. For opinions drawn from a set {1, 2, ..., k}, the opinion of the vertex would change by +1 if the opinion of the neighbour is larger, or by −1, if it is smaller.

We consider a population protocol version of the SIR model. In every round, an individual is chosen uniformly at random. If the individual is susceptible, then it becomes infected w.p. $\beta I_t/N$, where $I_t$ is the number of infections at time $t$ and $N$ is the total number of individuals. If the individual is infected, then it recovers w.p. $\gamma$, whereas, if the individual is already recovered, nothing happens. We prove sharp bounds on the probability of the disease becoming pandemic vs extinguishing early (dying out quickly). The probability of extinguishing early, $\Pr{\mathcal{E}_{ext}}$, is typically neglected in prior work since most use (deterministic) differential equations. Leveraging on this, using $\Pr{\mathcal{E}_{ext}}$, we proceed by bounding the expected size of the population that contracts the disease $\mathbf{E}\left[R_\infty\right]$. Prior work only calculated $\mathbf{E}\left[R_\infty | \overline{\mathcal{E}_{ext}}\right]$, or obtained non-closed form solutions. We then study the two-country model also accounting for the role of $\Pr{\mathcal{E}_{ext}}$. We assume that both countries have different infection rates $\beta^{(i)}$, but share the same recovery rate $\gamma$. In this model, each round has two steps: First, an individual is chosen u.a.r. and travels w.p. $p_{travel}$ to the other country. Afterwards, the process continues as before with the respective infection rates. Finally, using simulations, we characterise the influence of $p_{travel}$ on the total number of infections. Our simulations show that, depending on the $\beta^{(i)}$, increasing $p_{travel}$ can decrease or increase the expected total number of infections $\mathbf{E}\left[R_\infty\right]$.

We consider the rank of a class of sparse Boolean matrices of size $n \times n$. In particular, we show that the probability that such a matrix has full rank, and is thus invertible, is a positive constant with value about 0.2574 for large $n$. The matrices arise as the vertex-edge incidence matrix of 1-out 3-uniform hypergraphs. The result that the null space is bounded in expectation can be contrasted with results for the usual models of sparse Boolean matrices, based on the vertex-edge incidence matrix of random $k$-uniform hypergraphs. For this latter model, the expected co-rank is linear in the number of vertices $n$, [A. Coja-Oghlan et al., in Proceedings of SODA, 2020, pp. 579--591], [C. Cooper, A. M. Frieze, and W. Pegden, in Proceedings of SODA, 2019, pp. 946--955]. For fields of higher order, the co-rank is typically Poisson distributed.

We consider the rank of a class of sparse Boolean matrices of size $n \times n$. In particular, we show that the probability that such a matrix has full rank, and is thus invertible, is a positive constant with value about 0.2574 for large $n$. The matrices arise as the vertex-edge incidence matrix of 1-out 3-uniform hypergraphs. The result that the null space is bounded in expectation can be contrasted with results for the usual models of sparse Boolean matrices, based on the vertex-edge incidence matrix of random $k$-uniform hypergraphs. For this latter model, the expected co-rank is linear in the number of vertices $n$, [A. Coja-Oghlan et al., in Proceedings of SODA, 2020, pp. 579--591], [C. Cooper, A. M. Frieze, and W. Pegden, in Proceedings of SODA, 2019, pp. 946--955]. For fields of higher order, the co-rank is typically Poisson distributed.

We study a random walk that prefers tou se unvisited edges in the context of random cubic graphs. We establish asymptotically correct estimates for the vertex and edge cover times, these being $\approx n\log n$ and $\approx \frac32n\log n$ respectively.

We consider an asynchronous voting process on graphs called discordant voting, which can be described as follows. Initially each vertex holds one of two opinions, red or blue. Neighboring vertices with different opinions interact pairwise along an edge. After an interaction both vertices have the same color. The quantity of interest is the time to reach consensus, i.e., the number of steps needed for all vertices have the same color. We show that for a given initial coloring of the vertices, the expected time to reach consensus depends strongly on the underlying graph and the update rule (i.e., push, pull, oblivious).

We consider the model of population protocols, which can be viewed as a sequence of random pairwise interactions of n agents (nodes). During each interaction, two agents v and w selected uniformly at random update their states on the basis of their current states, and the whole system should in long run converge towards a desired global final configuration. We study population protocols for two problems: the leader election and the exact majority voting. Both protocols use Θ(log2 n) states per agent and run in O(log2 n) rounds (the number of interactions divided by n), w.h.p. and in expectation, improving on the running time of the Θ(log2 n)-state protocols proposed recently by Alistarh et al. [SODA 2017]. Our protocols are based on the idea of agents counting their local interactions and rely on the probabilistic fact that the uniform random selection would limit the divergence of the individual counts.

We present improved bounds on the cover time of the coalescing-branching random walk process COBRA. The COBRA process, introduced in [Dutta et al., SPAA 2013], can be viewed as spreading a single item of information throughout an undirected graph in synchronised rounds. In each round, each vertex which has received the information in the previous round (possibly simultaneously from more than one neighbour and possibly not for the first time), 'pushes' the information to b randomly selected neighbours. The COBRA process is typically studied for integer branching rates b \ge 2 (with the case b=1 corresponding to a random walk). The aim of the process is to propagate the information quickly, but with a limited number of transmissions per vertex per round.

In a voting process on a graph vertices revise their opinions in a distributed way based on the opinions of nearby vertices. The voting completes when the vertices reach consensus, that is, they all have the same opinion. The classic example is synchronous pull voting where at each step, each vertex adopts the opinion of a random neighbour. This very simple process, however, can be slow and the final opinion is not necessarily the one with the initial largest support. It was shown earlier that if there are initially only two opposing opinions, then both these drawbacks can be overcome by a synchronous two-sample voting, in which at each step each vertex considers its own opinion and the opinions of two random neighbours. If there are initially three or more opinions, a problem arises when there is no clear majority. One class of opinions may be largest (the plurality opinion), although its total size is less than that of two other opinions put together. We analyse the performance of the two-sample voting on d-regular graphs for this case. We show that, if the difference between the initial sizes A_1 and A_2 of the largest and second largest opinions is at least C n max{sqrt((log n)/A_1), lambda}, then the largest opinion wins in O((n log n)/A_1) steps with high probability. Here C is a suitable constant and lambda is the absolute second eigenvalue of transition matrix P=Adj(G)/d of a simple random walk on the graph G. Our bound generalizes the results of Becchetti et al. [SPAA 2014] for the related three-sample voting process on complete graphs. Our bound implies that if lambda = o(1), then the two-sample voting can consistently converge to the largest opinion, even if A_1 - A_2 = o(n). If lambda is constant, we show that the case A_1 - A_2 = o(n) can be dealt with by sampling using short random walks. Finally, we give a simple and efficient push voting algorithm for the case when there are a number of large opinions and any of them is acceptable as the final winning opinion.

We consider an asynchronous voting process on graphs which we call discordant voting, and which can be described as follows. Initially each vertex holds one of two opinions, red or blue say. Neighbouring vertices with different opinions interact pairwise. After an interaction both vertices have the same colour. The quantity of interest is T, the time to reach consensus, i.e. the number of interactions needed for all vertices have the same colour. An edge whose endpoint colours differ (i.e. one vertex is coloured red and the other one blue) is said to be discordant. A vertex is discordant if its is incident with a discordant edge. In discordant voting, all interactions are based on discordant edges. Because the voting process is asynchronous there are several ways to update the colours of the interacting vertices. - Push: Pick a random discordant vertex and push its colour to a random discordant neighbour. - Pull: Pick a random discordant vertex and pull the colour of a random discordant neighbour. - Oblivious: Pick a random endpoint of a random discordant edge and push the colour to the other end point. We show that ET, the expected time to reach consensus, depends strongly on the underlying graph and the update rule. For connected graphs on n vertices, and an initial half red, half blue colouring the following hold. For oblivious voting, ET = (n^2)/4 independent of the underlying graph. For the complete graph Kn, the push protocol has ET = Theta(n*log(n)), whereas the pull protocol has ET = Theta(2^n). For the cycle C_n all three protocols have ET = Theta(n^2). For the star graph however, the pull protocol has ET = O(n^2), whereas the push protocol is slower with ET = Theta(n^2*log(n)). The wide variation in ET for the pull protocol is to be contrasted with the well known model of synchronous pull voting, for which ET = O(n) on many classes of expanders.

We study voting models on graphs. In the beginning, the vertices of a given graph have some initial opinion. Over time, the opinions on the vertices change by interactions between graph neighbours. Under suitable conditions the system evolves to a state in which all vertices have the same opinion. In this work, we consider a new model of voting, called the Linear Voting Model. This model can be seen as a generalization of several models of voting, including among others, pull voting and push voting. One advantage of our model is that, even though it is very general, it has a rich structure making the analysis tractable. In particular we are able to solve the basic question about voting, the probability that certain opinion wins the poll, and furthermore, given appropriate conditions, we are able to bound the expected time until some opinion wins.

Information propagation on graphs is a fundamental topic in distributed computing. One of the simplest models of information propagation is the push protocol in which at each round each agent independently pushes the current knowledge to a random neighbour. In this paper we study the so-called coalescing-branching random walk (COBRA), in which each vertex pushes the information to k randomly selected neighbours and then stops passing information until it receives the information again. The aim of COBRA is to propagate information fast but with a limited number of transmissions per vertex per step. In this paper we study the cover time of the COBRA process defined as the minimum time until each vertex has received the information at least once. Our main result says that if G is an n-vertex r-regular graph whose transition matrix has second eigenvalue λ, then the COBRA cover time of G is O(log n), if 1-λ is greater than a positive constant, and O((log n)/(1-λ)3)), if 1-λ >> √log (i>n)/n}. These bounds are independent of r and hold for 3 ≤ r ≤ n-1. They improve the previous bound of O(log2 n) for expander graphs [Dutta et al., SPAA 2013].

Given a discrete random walk on a finite graph $G$, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step $t$. Let $\Gamma(t)$ be the subgraph of $G$ induced by the vacant set of the walk at step $t$. Similarly, let $\widehat \Gamma(t)$ be the subgraph of $G$ induced by the edges of the vacant net. For random $r$-regular graphs $G_r$, it was previously established that for a simple random walk the graph $\Gamma(t)$ of the vacant set undergoes a phase transition in the sense of the phase transition on Erdös--Renyi graphs $G_{n,p}$. Thus, for $r \ge 3$ there is an explicit value $t^*=t^*(r)$ of the walk such that for $t\leq (1-\epsilon)t^*$, $\Gamma(t)$ has a unique giant component, plus components of size $O(\log n)$, whereas for $t\geq (1+\epsilon)t^*$ all the components of $\Gamma(t)$ are of size $O(\log n)$. In this paper we establish the threshold value $\widehat t$ for a phase transition in the graph $\widehat \Gamma(t)$ of the vacant net of a simple random walk on a random $r$-regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a nonbacktracking random walk and, for $r$ even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random $r$-regular graphs. The main findings are the following: As $r$ increases, the threshold for the vacant set converges to $n \log r$ in all three walks. For the vacant net, the threshold converges to $rn/2 \; \log n$ for both the simple random walk and the nonbacktracking random walk. When $r\ge 4$ is even, the threshold for the vacant net of the unvisited edge process converges to $rn/2$, which is also the vertex cover time of the process.

In a coalescing random walk, a set of particles make independent discrete-time random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues a random walk through the graph. Let $G=(V,E)$ be an undirected and connected graph with $n$ vertices and $m$ edges. The coalescence time, $C(n)$, is the expected time for all particles to coalesce, when initially one particle is located at each vertex. We study the problem of bounding the coalescence time for general connected graphs and prove that $C(n) = O\big(\frac{1}{1-\lambda_2}\big(\log^{4} n + \frac{n}{\nu}\big)\big)$. Here $\lambda_2$ is the second eigenvalue of the transition matrix of the random walk. To avoid problems arising from, e.g., lack of coalescence on bipartite graphs, we assume the random walk can be made lazy if required. The value of $\nu$ is given by $\nu= \sum_{v\in V} d^2(v)/(d^2n)$, where $d(v)$ is the degree of vertex $v$, and $d=2m/n$ is the average degree. The parameter $\nu$ is an indicator of the variability of vertex degrees: $1 \le \nu = O(n)$, with $\nu=1$ for regular graphs. Our general bound on $C(n)$ holds for all connected graphs. This implies, for example, that $C(n)=O(n/(1-\lambda_2))$ for $d$-regular graphs with expansion parameterized by the eigenvalue gap $1-\lambda_2$. The bound on $C(n)$ given above is sublinear for some classes of graphs with skewed degree distributions. In the voter model, initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbor. Let ${\mathbf{E}} (C_{{\mbox{\boldmath$v$}}})$ be the expected time for voting to complete, that is, for a unique opinion to emerge. A system of coalescing particles, where initially one particle is located at each vertex, corresponds to the voter model in that $\mathbf{E}(C_{{\mbox{\boldmath$v$}}})=C(n)$. Thus our result stated above for $C(n)$ also gives general bounds for $\mathbf{E}(C_{{\mbox{\boldmath$v$}}})$.

In this paper, we consider a modified random walk which uses unvisited edges whenever possible, and makes a simple random walk otherwise. We call such a walk an edge-process (or E-process). We assume there is a rule A, which tells the walk which unvisited edge to use whenever there are several unvisited edges. In the simplest case, A is a uniform random choice over unvisited edges incident with the current walk position. However we do not exclude arbitrary choices of rule A. For example, the rule could be determined on-line by an adversary, or could vary from vertex to vertex.

In a coalescing random walk, a set of particles make independent discrete-time random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph. Coalescing random walks can be used to achieve consensus in distributed networks, and is the basis of the self-stabilizing mutual exclusion algorithm of Israeli and Jalfon [14].

We present a new randomized diffusion-based algorithm for balancing indivisible tasks (tokens) on a network. Our aim is to minimize the discrepancy between the maximum and minimum load. The algorithm works as follows. Every vertex distributes its tokens as evenly as possible among its neighbors and itself. If this is not possible without splitting some tokens, the vertex redistributes its excess tokens among all its neighbors randomly (without replacement). In this paper we prove several upper bounds on the load discrepancy for general networks. These bounds depend on some expansion properties of the network, that is, the second largest eigenvalue, and a novel measure which we refer to as refined local divergence. We then apply these general bounds to obtain results for some specific networks. For constant-degree expanders and torus graphs, these yield exponential improvements on the discrepancy bounds compared to the algorithm of Rabani, Sinclair, and Wanka [14]. For hypercubes we obtain a polynomial improvement. In contrast to previous papers, our algorithm is vertex-based and not edge-based. This means excess tokens are assigned to vertices instead to edges, and the vertex reallocates all of its excess tokens by itself. This approach avoids nodes having “negative loads” (like in [8, 10]), but causes additional dependencies for the analysis.

We present a family of peer-to-peer network protocols that yield regular graph topologies having known Hamilton cycles. These topologics are equivalent, in a well-defined sense, to the random regular graph. As a consequence, we have connectivity deterministically, and logarithmic diameter and expansion properties with high probability. We study the efficacy of certain simple topology-altering operations, designed to introduce randomness. These operations enable the network to self-stabilise when damaged. They resemble the operations used by Cooper, Dyer and Greenhill (2007) for a similar purpose in the case of random regular graphs. There is a link between our protocols and certain combinatorial structures which have been studied previously, in particular discordant permutations and Latin rectangles. We give the first rigorous polynomial mixing-time bounds for natural Markov chains that sample these objects at random. We do so by developing a novel extension to the canonical path technique for bounding mixing times: routing via a random destination. This resembles a technique used by Valiant (1982) for low-congestion routing in hypercubes.

We consider random walks on two classes of random graphs and explore the likely structure of the the set of unvisited vertices (or vacant set). Let Γ(t) be the subgraph induced by the vacant set. We show that for random graphs Gn,p above the connectivity threshold, and for random regular graphs Gr, for constant r ≥ 3, there is a phase transition in the sense of the well-known Erdős-Renyi phase transition. Thus for t ≤ (1 − ∊)t* we have a unique giant plus components of size O(log n) and for t ≥ (1 + ∊)t* we have only components of size O(log n). In the case of Gr we describe the likely degree sequence and structure of the small (O(log n)) size components.

We consider the following marking process (rw-rand) made by a random walk on an undirected graph G. Upon arrival at a vertex v, it marks v if unmarked and otherwise it marks a randomly chosen unmarked neighbor of v. We also consider a variant of this process called rw-r-rank. Here each vertex is assigned a global random rank first and then in each step, the walk marks the lowest ranked unmarked neighbor of the currently visited vertex. Depending on the degree and the expansion of the graph, we prove several upper bounds on the time required by these processes to mark all vertices. For instance, if G is a hypercube or random graph, our processes mark all vertices in time O(n), significantly speeding up the Θ(n log n)-cover time of standard random walks.

Let ${\bf A}$ be a $0/1$ matrix of size $m\times n$, and let p be the density of ${\bf A}$ (i.e., the number of ones divided by $m\cdot n$). We show that ${\bf A}$ can be approximated in the cut norm within $\varepsilon\cdot mnp$ by a sum of cut matrices (of rank 1), where the number of summands is independent of the size $m\cdot n$ of ${\bf A}$, provided that ${\bf A}$ satisfies a certain boundedness condition. This decomposition can be computed in polynomial time. This result extends the work of Frieze and Kannan [Combinatorica, 19 (1999), pp. 175–220] to sparse matrices. As an application, we obtain efficient $1-\varepsilon$ approximation algorithms for “bounded” instances of MAX CSP problems.

If $m\geq2$ is constant and $0\leq r\leq\varepsilon\log\log n$ for a small positive constant $\varepsilon$, then whp a random walk with look-ahead r on a scale-free graph $G=G_{(m,n)}$ has cover time $C_G(r)\sim(2/(m^{r-1}(m-1)))\;n\log n$.

Let $\mathbf{d}=d_1\leq d_2\leq\dots\leq d_n$ be a nondecreasing sequence of n positive integers whose sum is even. Let $\mathcal{G}_{n,\mathbf{d}}$ denote the set of graphs with vertex set $[n]=\{1,2,\dots,n\}$ in which the degree of vertex i is $d_i$. Let $G_{n,\mathbf{d}}$ be chosen uniformly at random from $\mathcal{G}_{n,\mathbf{d}}$. It will be apparent from section 4.3 that all of the sequences we are considering will be graphic. We give a condition on $\mathbf{d}$ under which we can show that whp $\mathcal{G}_{n,\mathbf{d}}$ is Hamiltonian. This condition is satisfied by graphs with exponential tails as well those with power law tails.

We define a network that relies on its protocol's emergent behaviour to maintain the useful properties of a random regular topology. It does this by spontaneously performing flips in an effort to randomise [15], allowing it to repair damage and to embed new peers without over-complicated joining schema.

Let A be a 0/1 matrix of size m×n, and let p be the density of A (i.e., the number of ones divided by m · n). We show that A can be approximated in the cut norm within ∊ · mnp by a sum of cut matrices (of rank 1), where the number of summands is independent of the size m · n of A, provided that A satisfies a certain boundedness condition. The decomposition can be computed in polynomial time. This result extends the work of Frieze and Kannan (Combinatorica 1999) to sparse matrices. As an application, we obtain efficient 1 – ∊ approximation algorithms for “bounded” instances of Max CSP problems.

We study the cover time of random geometric graphs. Let I(d) = [0, 1]d denote the unit torus in d dimensions. Let D(x, r) denote the ball (disc) of radius r. Let ϒd be the volume of the unit ball D(0, 1) in d dimensions. A random geometric graph G = G(d, r, n) in d dimensions is defined as follows: Sample n points V independently and uniformly at random from I(d). For each point x draw a ball D(x, r) of radius r about x. The vertex set V(G) = V and the edge set E (G) = {{v, w} : w ≠ v, w ∊ D(v, r)}. Let G(d, r, n), d ≥ 3 be a random geometric graph. Let c > 1 be constant, and let r = (c log n/(ϒdn))1/d. Then whp

We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make the analysis for random regular graphs. The cover time of a random walk on a random r-regular graph was studied in [C. Cooper and A. Frieze, SIAM J. Discrete Math., 18 (2005), pp. 728–740], where it was shown with high probability (whp) that for $r\geq3$ the cover time is asymptotic to $\theta_r n\ln n$, where $\theta_r=(r-1)/(r-2)$. In this paper we prove the following (whp) results, arising from the study of multiple random walks on a random regular graph G. For k independent walks on G, the cover time $C_G(k)$ is asymptotic to $C_G/k$, where $C_G$ is the cover time of a single walk. For most starting positions, the expected number of steps before any of the walks meet is $\theta_r n/\binom{k}{2}$. If the walks can communicate when meeting at a vertex, we show that, for most starting positions, the expected time for k walks to broadcast a single piece of information to each other is asymptotic to $\frac{2\ln k}{k}\theta_r n$ as $k,n\rightarrow\infty$. We also establish properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing or by annihilating each other. For example, the expected extinction time of k explosive particles (k even) tends to $(2\ln2)\theta_r n$ as $k\rightarrow\infty$. The case of n coalescing particles, where one particle is initially located at each vertex, corresponds to a voter model defined as follows: Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbor. The expected time for a unique opinion to emerge is the same as the expected time for all the particles to coalesce, which is asymptotic to $2\theta_r n$. Combining results from the predator-prey and multiple random walk models allows us to compare expected detection times of all prey in the following scenarios: Both the predator and the prey move randomly, the prey moves randomly and the predators stay fixed, and the predators move randomly and the prey stays fixed. In all cases, with k predators and $\ell$ prey the expected detection time is $\theta_r H_{\ell}n/k$, where $H_{\ell}$ is the $\ell$th harmonic number.

This paper studies broadcasting and gossiping algorithms in random and general AdHoc networks. Our goal is not only to minimise the broadcasting and gossiping time, but also to minimise the energy consumption, which is measured in terms of the total number of messages (or transmissions) sent. We assume that the nodes of the network do not know the network, and that they can only send with a fixed power, meaning they can not adjust the area sizes that their messages cover. We believe that under these circumstances the number of transmissions is a very good measure for the overall energy consumption.

Let $r \ge 3$ be constant, and let ${\cal G}_{r}$ denote the set of r-regular graphs with vertex set V = {1,2,...,n}. Let G be chosen randomly from ${\cal G}_{r}$. We prove that with high probability (\whp) the cover time of a random walk on G is asymptotic to $\frac{r-1}{r-2}\;n\log n$.

Introduction We consider a simple model of an agent (which we call a spider) moving between the nodes of a randomly growing web graph. It is presumed that the agent examines the page content of the node for some specific topic. In our model the spider makes a random walk on the existing set of vertices. We compare the success of the spider on web graphs of two distinct types. For a random graph web graph model, in which new vertices join edges to existing vertices uniformly at random, the expected proportion of unvisited vertices tends to 0.57. For the comparable copy-based web graph model, in which new vertices join edges to existing vertices proportional to vertex degree, the expected proportion of unvisited vertices tends to 0.59. A web graph is a sparse connected graph designed to capture some properties of the www. Studies of the graph structure of the www were made by [4] and [7] among others. There are many models of web graphs designed to capture the structure of the www foun