## Section: New Results

### Network and Graph Algorithms

#### Vertex Coloring with Communication and Local Memory Constraints in Synchronous Broadcast Networks

Participants : Hicham Lakhlef, Michel Raynal, Francois Taiani.

This work [41] considers the broadcast/receive communication model in which message collisions and message conflicts can occur because processes share frequency bands. (A collision occurs when, during the same round, messages are sent to the same process by too many neighbors. A conflict occurs when a process and one of its neighbors broadcast during the same round.) More precisely, this work considers the case where, during a round, a process may either broadcast a message to its neighbors or receive a message from at most $m$ of them. This captures communication-related constraints or a local memory constraint stating that, whatever the number of neighbors of a process, its local memory allows it to receive and store at most $m$ messages during each round. This work defines first the corresponding generic vertex multi-coloring problem (a vertex can have several colors). It focuses then on tree networks, for which it presents a lower bound on the number of colors $K$ that are necessary (namely, $K=\lceil \frac{\Delta}{m}\rceil +1$, where $\Delta $ is the maximal degree of the communication graph), and an associated coloring algorithm, which is optimal with respect to $K$.

#### Optimal Collision/Conflict-Free Distance-2 Coloring in Wireless Synchronous Broadcast/Receive Tree Networks

Participants : Davide Frey, Hicham Lakhlef, Michel Raynal.

We studied the problem of decentralized distance-2 coloring in message-passing systems where communication is (a) synchronous and (b) based on the “broadcast/receive” pair of communication operations. “Synchronous” means that time is discrete and appears as a sequence of time slots (or rounds) such that each message is received in the very same round in which it is sent. “Broadcast/receive” means that during a round a process can either broadcast a message to its neighbors or receive a message from one of them. In such a communication model, no two neighbors of the same process, nor a process and any of its neighbors, must be allowed to broadcast during the same time slot (thereby preventing message collisions in the first case, and message conflicts in the second case). From a graph theory point of view, the allocation of slots to processes is know as the distance-2 coloring problem: a color must be associated with each process (defining the time slots in which it will be allowed to broadcast) in such a way that any two processes at distance at most 2 obtain different colors, while the total number of colors is “as small as possible”. In this context, we proposed a parallel message-passing distance-2 coloring algorithm suited to trees, whose roots are dynamically defined. This algorithm, which is itself collision-free and conflict-free, uses $\Delta +1$ colors where $\Delta $ is the maximal degree of the graph (hence the algorithm is color-optimal). It does not require all processes to have different initial identities, and its time complexity is $O\left(d\Delta \right)$, where $d$ is the depth of the tree. As far as we know, this is the first distributed distance-2 coloring algorithm designed for the broadcast/receive round-based communication model, which owns all the previous properties. We published these results in [39].

#### Efficient Plurality Consensus, or: The Benefits of Cleaning Up from Time to Time

Participant : George Giakkoupis.

Plurality consensus considers a network of $n$ nodes, each having one of $k$ opinions.
Nodes execute a (randomized) distributed protocol with the goal that all nodes adopt the *plurality* (the opinion initially supported by the most nodes).
Communication is realized via the random phone call model.
A major open question has been whether there is a protocol for the complete graph that converges (w.h.p.) in polylogarithmic time and uses only polylogarithmic memory per node (local memory).
We answered this question affirmatively.

In [22], we propose two protocols that need only mild assumptions on the bias in favor of the plurality.
As an example of our results, consider the complete graph and an arbitrarily small constant multiplicative bias in favor of the plurality.
Our first protocol achieves plurality consensus in $O(logk\xb7loglogn)$ rounds using $logk+O(loglogk)$ bits of local memory.
Our second protocol achieves plurality consensus in $O(logn\xb7loglogn)$ rounds using only $logk+4$ bits of local memory.
This disproves a conjecture by Becchetti et al. (SODA'15) implying that any protocol with local memory $logk+O\left(1\right)$ has worst-case runtime $\Omega \left(k\right)$.
We provide similar bounds for much weaker bias assumptions.
At the heart of our protocols lies an *undecided state*, an idea introduced by Angluin et al. (Distributed Computing'08).

This work was done in collaboration with Petra Berenbrink (SFU), Tom Friedetzky (Durham University), and Peter Kling (SFU).

#### Bounds on the Voter Model in Dynamic Networks

Participants : George Giakkoupis, Anne-Marie Kermarrec.

In the *voter model*, each node of a graph has an opinion, and in every round each node chooses independently a random neighbour and adopts its opinion. We are interested in the *consensus time*, which is the first point in time where all nodes have the same opinion.
In [23], we consider dynamic graphs
in which the edges are rewired in every round (by an adversary) giving rise to the graph sequence ${G}_{1},{G}_{2},\cdots $, where we assume that ${G}_{i}$ has conductance at least ${\phi}_{i}$.
We assume that the degrees of nodes don't change over time as one can show that the consensus time can become super-exponential otherwise.
In the case of a sequence of $d$-regular graphs, we obtain asymptotically tight results.
Even for some static graphs, such as the cycle, our results improve the state of the art.
Here we show that the expected number of rounds until all nodes have the same opinion is bounded by $O(m/(\delta \xb7\phi \left)\right)$, for any graph with $m$ edges, conductance $\phi $, and degrees at least $\delta $.
In addition, we consider a *biased* dynamic voter model, where each opinion $i$ is associated with a probability ${P}_{i}$, and when a node chooses a neighbour with that opinion, it adopts opinion $i$ with probability ${P}_{i}$ (otherwise the node keeps its current opinion).
We show for any regular dynamic graph, that if there is an $\u03f5>0$ difference between the highest and second highest opinion probabilities, and at least $\Omega (logn)$ nodes have initially the opinion with the highest probability, then all nodes adopt w.h.p. that opinion.
We obtain a bound on the convergence time, which becomes $O(logn/\phi )$ for static graphs.

This work was done in collaboration with Petra Berenbrink (SFU), and Frederik Mallmann-Trenn (SFU).

#### How Asynchrony Affects Rumor Spreading Time

Participant : George Giakkoupis.

In standard randomized (push-pull) rumor spreading, nodes communicate in synchronized rounds. In each round every node contacts a random neighbor in order to exchange the rumor (i.e., either push the rumor to its neighbor or pull it from the neighbor). A natural asynchronous variant of this algorithm is one where each node has an independent Poisson clock with rate 1, and every node contacts a random neighbor whenever its clock ticks. This asynchronous variant is arguably a more realistic model in various settings, including message broadcasting in communication networks, and information dissemination in social networks.

In [35] we study how asynchrony affects the rumor spreading time, that is, the time before a rumor originated at a single node spreads to all nodes in the graph. Our first result states that the asynchronous push-pull rumor spreading time is asymptotically bounded by the standard synchronous time. Precisely, we show that for any graph $G$ on $n$-nodes, where the synchronous push-pull protocol informs all nodes within $T\left(G\right)$ rounds with high probability, the asynchronous protocol needs at most time $O\left(T\right(G)+logn)$ to inform all nodes with high probability. On the other hand, we show that the expected synchronous push-pull rumor spreading time is bounded by $O\left(\sqrt{n}\right)$ times the expected asynchronous time.

These results improve upon the bounds for both directions shown recently by Acan et al. (PODC 2015). An interesting implication of our first result is that in regular graphs, the weaker push-only variant of synchronous rumor spreading has the same asymptotic performance as the synchronous push-pull algorithm.

This work was done in collaboration with Yasamin Nazari and Philipp Woelfel from the University of Calgary.

#### Amplifiers and Suppressors of Selection for the Moran Process on Undirected Graphs

Participant : George Giakkoupis.

In [47] we consider the classic Moran process modeling the spread of genetic mutations, as extended to structured populations by Lieberman et al. (Nature, 2005). In this process, individuals are the vertices of a connected graph $G$. Initially, there is a single mutant vertex, chosen uniformly at random. In each step, a random vertex is selected for reproduction with a probability proportional to its fitness: mutants have fitness $r>1$, while non-mutants have fitness 1. The vertex chosen to reproduce places a copy of itself to a uniformly random neighbor in $G$, replacing the individual that was there. The process ends when the mutation either reaches fixation (i.e., all vertices are mutants), or gets extinct. The principal quantity of interest is the probability with which each of the two outcomes occurs.

A problem that has received significant attention recently concerns the existence of families of graphs, called strong amplifiers of selection, for which the fixation probability tends to 1 as the order $n$ of the graph increases, and the existence of strong suppressors of selection, for which this probability tends to 0. For the case of directed graphs, it is known that both strong amplifiers and suppressors exist. For the case of undirected graphs, however, the problem has remained open, and the general belief has been that neither strong amplifiers nor suppressors exist. In this work we disprove this belief, by providing the first examples of such graphs. The strong amplifier we present has fixation probability $1-\tilde{O}\left({n}^{-1/3}\right)$, and the strong suppressor has fixation probability $\tilde{O}\left({n}^{-1/4}\right)$. Both graph constructions are surprisingly simple. We also prove a general upper bound of $1-\tilde{\Omega}\left({n}^{-1/3}\right)$ on the fixation probability of any undirected graph. Hence, our strong amplifier is existentially optimal.