We study the problem of energy-balanced data propagation in wireless sensor networks. The energy balance property guarantees that the average per sensor energy dissipation is the same for all sensors in the network, during the entire execution of the data propagation protocol. This property is important since it prolongs the network's lifetime by avoiding early energy depletion of sensors.We propose a new algorithm that in each step decides whether to propagate data one-hop towards the final destination (the sink), or to send data directly to the sink. This randomized choice balances the (cheap) onehop transimssions with the direct transimissions to the sink, which are more expensive but "bypass" the sensors lying close to the sink. Note that, in most protocols, these close to the sink sensors tend to be overused and die out early.By a detailed analysis we precisely estimate the probabilities for each propagation choice in order to guarantee energy balance. The needed estimation can easily be performed by current sensors using simple to obtain information. Under * some assumptions, we also derive a closed form for these probabilities.The fact (shown by our analysis) that direct (expensive) transmissions to the sink are needed only rarely, shows that our protocol, besides energy-balanced, is also energy efficient.

The independence number of a sparse random graph G(n, m) of average degree d = 2m/n is well-known to bewith high probability, with ε d → 0 in the limit of large d. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size n ln(d)/d, i.e., about half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with (1 + ε)n ln(d)/d for any fixed ε > 0 (independent of both d and n). In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size k of the independent sets passes the point k ∼ n ln(d)/d. Roughly speaking, we prove that independent sets of size k > (1 + ε)n ln(d)/d form an intricately ragged landscape, in which local search algorithms seem to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independents sets.

Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph. Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science. In a path-breaking paper based on the non-rigorous 'cavity method', physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, computer science and physics [Krzakala et al.: PNAS 2007]. In this paper we rigorise this picture completely for a broad class of models, encompassing the Potts antiferromagnet on the random graph, the k-XORSAT model and the diluted k-spin model for even k. We also prove a conjecture about the detection problem in the stochastic block model that has received considerable attention [Decelle et al.: Phys. Rev. E 2011].1 k h=1 ρ π h (σ h ) . 2 and d cond (k, β) = inf{d > 0 : sup π∈P 2 * ({1,−1}) B k−spin (d, β, π) > ln 2}. Then 0 < d cond (k, β) < ∞ and (k, β).From now on we assume that k ≥ 4 is even. The regime d < d cond (k, β) is called the replica symmetric phase. According to the cavity method, its key feature is that with probability tending to 1 in the limit n → ∞, two independent samples σ 1 , σ 2 ('replicas') chosen from the Gibbs measure µ H,J ,β are "essentially perpendicular". To formalize this define for σ, τ : V n → {±1} the overlap as ̺ σ,τ = x∈V n σ(x)τ(x)/n. We write 〈 · 〉 H,J ,β for the average on σ 1 , σ 2 chosen independently from µ H,J ,β and denote the expectation over the choice of H and J by E [ · ]. Theorem 1.2. For all β > 0 and k ≥ 4 even we have d cond (k, β) The corresponding statement for k = 2 was proved by Guerra and Toninelli, but as they point out their argument does not extend to larger k [37]. Theorem 1.2 implies the absence of extensive long-range correlations in the replica symmetric phase. Indeed, for two vertices x, y ∈ V n and s, t ∈ {+1, −1} letbe the joint distribution of the spins assigned to x, y. Further, letρ be the uniform distribution on {±1} × {±1}. Then the total variation distance µ H,J ,β,x,y −ρ TV is a measure of how correlated the spins of x, y are. Indeed, in the case that k is even for every x ∈ V n the Gibbs marginals satisfy µ H,J ,β,x (±1) = 〈1{σ 1 (x) = ±1}〉 H,J ,β = 1/2 because µ H,J ,β (σ) = µ H,J ,β (−σ) for every σ ∈ {−1, +1} n . Therefore, if the spins at x, y were independent, then µ H,J ,β,x,y = µ H,J ,β,x ⊗ µ H,J ,β,y =ρ. Furthermore, it is well known (e.g., [13, Section 2]) thatThus, Theorem 1.2 implies that for d < d cond (k, β), with probability tending to 1, the spins assigned to two random vertices x, y of H are asymptotically independent. By contrast, Theorem 1.2 and (1.2) show that extensive longrange dependencies occur beyond but arbitrarily close to d cond (k, β).1.3. The Potts antiferromagnet. Let q ≥ 2 be an integer, let Ω = {1, . . . , q} be a set...

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